A critical non-homogeneous heat equation with weighted source

TL;DR

This paper investigates the qualitative behavior of radially symmetric solutions to a critical non-homogeneous heat equation with weighted sources, identifying conditions for blow-up or decay, and exploring solutions in the limiting case.

Contribution

It provides new insights into the behavior of solutions to a weighted heat equation with critical density, including existence, decay rates, and large-time behavior, using a transformation to a Fisher-KPP type equation.

Findings

Conditions for finite-time blow-up or decay of solutions.

Existence of non-trivial solutions in the critical case.

Decay rates and asymptotic behavior of solutions.

Abstract

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=\Delta u+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are obtained, in the range of exponents $p>1$, $\sigma\ge-2$. More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as $t\to\infty$ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case $\sigma=-2$ we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher-KPP equation is derived and employed in order to deduce these properties.